Section 1.4 A Sense of Scale: Significant Figures, Scientific Notation, and Units © 2015 Pearson Education, Inc

Section 1.4 A Sense of Scale:Significant Figures, Scientific Notation,

and Units

© 2015 Pearson Education, Inc.

Slide 1-2

Measurements and Significant Figures

• When we measure any quantity we can do so with only a certain precision.

• We state our knowledge of a measurement through the use of significant figures: digits that are reliably known.


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Text: p. 12

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QuickCheck 1.7

Rank in order, from the most to the least, the number of significant figures in the following numbers. For example, if b has more than c, c has the same number as a, and a has more than d, you would give your answer as b > c = a > d.

a. 8200 b. 0.0052 c. 0.430 d. 4.321 × 10–10

A. d > c > b = aB. a = b = d > cC. b = d > c > aD. d > c > a > bE. a = d > c > b


Slide 1-5

Scientific Notation

• Writing very large (much greater than 1) and very small (much less than 1) numbers is cumbersome and does not make clear how many significant figures are involved.


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Text: p. 13

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Units

• Scientists use a system of units called le Système International d’Unités, commonly referred to as SI Units.


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Unit Conversions


Text: p. 15

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Estimation

• A one-significant-figure estimate or calculation is called an order-of-magnitude estimate.

• An order-of-magnitude estimate is indicated by the symbol ~, which indicates even less precision than the “approximately equal” symbol ≈.


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Example 1.5 How fast do you walk?

Estimate how fast you walk, in meters per second.

PREPARE In order to compute speed, we need a distance and a time. If you walked a mile to campus, how long would this take? You’d probably say 30 minutes or so—half an hour. Let’s use this rough number in our estimate.


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Example 1.5 How fast do you walk? (cont.)

SOLVE Given this estimate, we compute your speed as

But we want the speed in meters per second. Since our calculation is only an estimate, we use an approximate conversion factor from Table 1.4:

This gives an approximate walking speed of 1 m/s.


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Example 1.5 How fast do you walk? (cont.)

ASSESS Is this a reasonable value? Let’s do another estimate. Your stride is probably about 1 yard long—about 1 meter. And you take about one step per second; next time you are walking, you can count and see. So a walking speed of 1 meter per second sounds pretty reasonable.


Section 1.5 Vectors and Motion:A First Look


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Scalars and Vectors

• When a physical quantity is described by a single number (with a unit), we call it a scalar quantity.

• A vector quantity is a quantity that has both a size (How far? or How fast?) and a direction (Which way?).

• The size or length of a vector is called its magnitude.

• We graphically represent a vector as an arrow.


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Displacement Vectors

• The displacement vector represents the distance and direction of an object’s motion.

• An object’s displacement vector is drawn from the object’s initial position to its final position, regardless of the actual path followed between these two points.


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Vector Addition

• The net displacement for a trip with two legs is the sum of the two displacements that made it up.


Text: p. 17

Slide 1-17

QuickCheck 1.6

Given vectors and , what is ?


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Example 1.7 How far away is Anna?

Anna walks 90 m due east and then 50 m due north. What is her displacement from her starting point?

PREPARE Let’s start with the sketch in FIGURE 1.25a. We set up a coordinate system with Anna’s original position as the origin, and then we drew her two subsequent motions as the two displacement vectors


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Example 1.7 How far away is Anna? (cont.)

SOLVE We drew the two vector displacements with the tail of one vector starting at the head of the previous one—exactly what is needed to form a vector sum. The vector net in FIGURE 1.25a is the vector sum of the successive displacements and thus represents Anna’s net displacement from the origin.


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Anna’s distance from the origin is the length of this vector net. FIGURE 1.25b shows that this vector is the hypotenuse of a right triangle with sides 50 m (because Anna walked 50 m north) and 90 m (because she walked 90 m east). We can compute the magnitude of this vector, her net displacement, using the Pythagorean theorem (the square of the length of the hypotenuse of a triangle is equal to the sum of the squares of the lengths of the sides):


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We have rounded off to the appropriate number of significant figures, giving us 100 m for the magnitude of the displacement vector. How about the direction? Figure 1.25b identifies the angle that gives the angle north of east of Anna’s displacement. In the right triangle, 50 m is the opposite side and 90 m is the adjacent side, so the angle is given by

Putting it all together, we get a net displacement of


Slide 1-22


ASSESS We can use our drawing to assess our result. If the two sides of the triangle are 50 m and 90 m, a length of 100 m for the hypotenuse seems about right. The angle is certainly less than 45°, but not too much less, so 29° seems reasonable.


Slide 1-23

Velocity Vectors

• We represent the velocity of an object by a velocity vector that points in the direction of the object’s motion, and whose magnitude is the object’s speed.

The motion diagram for a car starting from rest


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Example 1.8 Drawing a ball’s motion diagram

Jake hits a ball at a 60° angle from the horizontal. It is caught by Jim. Draw a motion diagram of the ball that shows velocity vectors rather than displacement vectors.

The motion diagram of a ball traveling from Jake to Jim


Section 1.6 Where Do We Go from Here?


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Summary and Organization of Chapters

• This chapter has been an introduction to some of the fundamental ideas about motion and some of the basic techniques that you will use.

• Each new chapter depends on those that preceded it.

• Each chapter begins with a chapter preview that will let you know which topics are especially important to review.

• The last element in each chapter will be an integrated example that brings together the principles and techniques you have just learned with those you learned previously.


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Integrated Example 1.9 A goose gets its bearings

FIGURE 1.28 shows the path of a Canada goose that flew in a straight line for some time before making a corrective right-angle turn. One hour after beginning, the goose made a rest stop on a lake due east of its original position.


Slide 1-28

Integrated Example 1.9 A goose gets its bearings (cont.)

a. How much extra distance did the goose travel due to its initial error in flight direction? That is, how much farther did it fly than if it had simply flown directly to its final position on the lake?

b. What was the flight speed of the goose?

c. A typical flight speed for a migrating goose is 80 km/h. Given this, does your result seem reasonable


Slide 1-29


Drawing and labeling the displacement between the starting and ending points in Figure 1.29 show that it is the hypotenuse of a right triangle, so we can use our rules for triangles as we look for a solution.


Slide 1-30


SOLVE

a. The minimum distance the goose could have flown, if it flew straight to the lake, is the hypotenuse of a triangle with sides 21 mi and 28 mi. This straight-line distance is

The actual distance the goose flew is the sum of the distances traveled for the two legs of the journey:

distance traveled = 21 mi + 28 mi = 49 mi

The extra distance flown is the difference between the actual distance flown and the straight-line distance—namely, 14 miles.


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SOLVE

b. To compute the flight speed, we need to consider the distance that the bird actually flew. The flight speed is the total distance flown divided by the total time of the flight:

c. To compare our calculated speed with a typical flight speed, we must convert our solution to km/h, rounding off to the correct number of significant digits:


Slide 1-32

Summary: Important Concepts


Text: p. 22

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Text: p. 22

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© 2015 Pearson Education, Inc. Text: p. 22

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Text: p. 22

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Summary: Applications


Text: p. 22

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Summary: Applications


Text: p. 22

Slide 1-38

Summary

© 2015 Pearson Education, Inc. Text: p. 22

Slide 1-39

Summary


Text: p. 22

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Section 1.4 A Sense of Scale: Significant Figures, Scientific Notation, and Units © 2015 Pearson Education, Inc